In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, Frobenius-and-add algorithm using the tau-adic nonadjacent form (tau-NAF). The tau-NAF has the three propertie...
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In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, Frobenius-and-add algorithm using the tau-adic nonadjacent form (tau-NAF). The tau-NAF has the three properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming weight. On the other hand, Gfinther et al. (Speeding up the arithmetic on koblitz curves of genus two. LNCS, vol. 2012, pp. 106-117. Springer, Heidelberg, 2001) have proposed two generalizations of tau-NAF for a family of hyperelliptic curves (hyperelliptic Koblitz curves) which have been proposed by Koblitz (J Cryptol 1(3):139-150, 1989). We call these generalizations tau-adic sparse expansion, and tau-NAF, respectively. To our knowledge, it is not known whether the three properties are true or not, especially, the existence must be satisfied for concrete cryptographic implementations. We provide an answer to the question. Our investigation shows that the tau-adic sparse expansion has only the existence and the tau-NAF has the existence and uniqueness. Our results guarantee the concrete cryptographic implementations of these generalizations.
We consider a multiple access relay channel (MARC) network consisting of two sources, one relay, and one common destination applying compute-and-forward (CF) strategy. We show that the direct application of CF to the ...
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ISBN:
(纸本)9781467389990
We consider a multiple access relay channel (MARC) network consisting of two sources, one relay, and one common destination applying compute-and-forward (CF) strategy. We show that the direct application of CF to the MARC network results in poor error performance bounded by (p + 1)(-1), the probability of rank deficiency of the coefficient matrix over F-p. To solve this problem, we propose two practical approaches. First, given an optimal coefficient vector at the relay, the destination is restricted to select a coefficient vector ensuring a full rank coefficient matrix. Second, given an optimal coefficient vector at the destination obtained via a small amount of feedback, the relay is restricted to choose a coefficient vector guaranteeing a full rank coefficient matrix. We simulate these CF implementation strategies using self-similar nested E-8 lattice codes and confirm that both of the proposed schemes outperform the direct implementation in terms of achievable transmission rate and frame-error-rate performance. Furthermore, we confirm that with a small amount of feedback, the second strategy is better than the first one. In addition, we present in detail a modified fincke-pohst algorithm for computing the coefficient candidates and show its efficiency compared to an exhaustive search.
In the present paper we show how to speed up lattice parameter searches for Monte Carlo and quasi-Monte Carlo node sets. The classical measure for such parameter searches is the spectral test which is based on a calcu...
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In the present paper we show how to speed up lattice parameter searches for Monte Carlo and quasi-Monte Carlo node sets. The classical measure for such parameter searches is the spectral test which is based on a calculation of the shortest nonzero vector in a lattice. Instead of the shortest vector we apply an approximation given by the LLL algorithm for lattice basis reduction. We empirically demonstrate the speed-up and the quality loss obtained by the LLL reduction, and we present important applications for parameter selections.
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